Note on formulation of the enhanced scattering- (transmittance-) matrix approach

The enhanced transmittance matrix approach developed by Moharam et al. [J. Opt. Soc. Am. A 12, 1077 (1995)] is reformulated in a concise and illuminating form in terms of scattering (reflection and transmission) matrices directly. Two equivalent recursive formulations, corresponding to their full- and partial-solution approaches, are presented and extended to allow simultaneous determination of both reflected and transmitted amplitudes. The relationships between these formulations and the S-matrix algorithm, together with their relative efficiencies and usefulness, are ascertained and compared by means of compact formulas featuring parallel algebraic structures. It is made evident that given the eigenmode solutions, the enhanced approach is the most direct and efficient way for deducing global scattering matrices.